Nuprl Lemma : surject-inverse
∀[A,B:Type]. ∀f:A ⟶ B. (Surj(A;B;f) ⇐⇒ ∃g:B ⟶ A. ∀x:B. ((f (g x)) = x ∈ B))
Proof
Definitions occuring in Statement :
surject: Surj(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
surject: Surj(A;B;f),
exists: ∃x:A. B[x],
pi1: fst(t),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
surject_wf,
exists_wf,
all_wf,
equal_wf,
pi1_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
functionEquality,
sqequalRule,
lambdaEquality,
universeEquality,
rename,
dependent_pairFormation,
productElimination,
dependent_pairEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
imageElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}f:A {}\mrightarrow{} B. (Surj(A;B;f) \mLeftarrow{}{}\mRightarrow{} \mexists{}g:B {}\mrightarrow{} A. \mforall{}x:B. ((f (g x)) = x))
Date html generated:
2017_04_17-AM-07_46_11
Last ObjectModification:
2017_02_27-PM-04_17_51
Theory : list_1
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